When you say “the power goes out” what do you mean? Are you saying your home constantly has the power turned off e.g power outages or you turn your circuit breakers off at night?
Most technology does not do well with constant power interruptions. It damages equipment like computers. Many systems or electronics in terms of their logic boards only remember things when there is some trickle of power. In the total absence of power the logic boards lose their settings. As you said “you would hope when turning the light off using the remote, the fan remembers the setting” yes for sure but if you’re having constant power outages or power flickers it’s not only going to screw up the logic / memory but eventually just fry the fan or other electronics in your home completely. I would ensure that if you are constantly losing power you look at having a UPS for some of the really important things in the home e.g your Internet modem, your computer, etc. I can guarantee you if you are turning circuit breakers off at night or you just have problems with power constantly being interrupted you will soon find some of your home electronics DO NOT turn back on after power is restored at all. E.G Microwave, dishwasher, fridge. Microwave will probably be the first to go, they really hate poor power regulation and constant spikes / brown outs.
If you’re having over loads that are causing circuits to flip off e.g you turn on an appliance and a circuit breaker flips off, then you need to look at why you’re overloading your circuits and distribute your electronics or appliances onto other circuits.
Might be time to call an electrician depending on what the scenario is.